The evaluation on identity gives the vertex amplitude (REF ).
To glue the vertex amplitudes associated to adjacent tetrahedra, we make use of the identity in the representation space \({\mathcal {V}}^j\otimes {\mathcal {V}}^{*j}\)  spanned by the coherent states [1]}
\(\mathbb {I}_{{\mathcal {V}}^j\otimes {\mathcal {V}}^{*j}}=\frac{1}{(2j)!}\int \mathrm {d}\mu (z) |j,z\rangle \langle j,z|=\frac{1}{(2j)!}\int \mathrm {d}\mu (z) |j,z][ j,z|\,.\) 
